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The statistical mechanics of systems in which the dominant process is a flow of energy through the modes of the system is studied. The case of randomly excited fluid turbulence is studied and it is argued that there is a strong mathematical analogy between the classical (turbulent) cascade of energy and the quantum field or many-body problem. The energy has an analogue in the one-particle Green function, and entropy can be defined, the latter being the information content in the case of the probability distribution function for the turbulence. General operations in Hilbert space can be carried out with at most two functions, and the energy equation and the maximization of the entropy give two equations which determine the two chosen unknown functions. The case of a random long-wave input of energy is studied and shown to lead to the Kolmogoroff spectrum, and the Kolmogoroff constant is evaluated for the approximation system used.
Edwards et al. (Sat,) studied this question.